AFIR explorations of transition states of extended unsaturated systems: automatic location of ambimodal transition states

Takuma Ito a, Yu Harabuchi *bcde and Satoshi Maeda *bcd
aGraduate School of Chemical Sciences and Engineering, Hokkaido University, Sapporo 060-8628, Japan
bDepartment of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan. E-mail: y_harabuchi@sci.hokudai.ac.jp
cInstitute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University, Sapporo 001-0021, Japan. E-mail: smaeda@eis.hokudai.ac.jp
dJST, ERATO Maeda Artificial Intelligence for Chemical Reaction Design and Discovery Project, Sapporo 060-0810, Japan
eJST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

Received 3rd May 2020 , Accepted 9th June 2020

First published on 10th June 2020


Paths of Diels–Alder reactions between 2-vinylfuran and 3-methoxycarbonylcyclopentadienone were systematically explored by the multicomponent version of the artificial force induced reaction (MC-AFIR) method. In this reaction, the dynamical bifurcation in which a single transition state (TS) relates to two different products has been reported to occur [J. B. Thomas, et al., J. Am. Chem. Soc., 2008, 130, 14544–14555]. In this paper, based on the MC-AFIR method, we propose a procedure to systematically explore so called ambimodal TSs through which the dynamical bifurcation occurs. The present procedure finds candidates of TSs that may cause the dynamical bifurcation from the logs of an automated reaction path search by the MC-AFIR method, without any additional quantum chemical calculations. For this reaction, the MC-AFIR search found 125 unique TSs automatically. Among the 125 TSs, 19 were suggested as candidates, and finally, six including the one reported in the literature were confirmed to cause the dynamical bifurcation. The present procedure would be promising to find TSs involved in the dynamical bifurcation automatically.


I. Introduction

The Diels–Alder reaction is one of the most important synthesis methods and has been utilized in vast chemical syntheses.1,2 Its transition state (TS) has also been studied extensively based on quantum chemical calculations.3 On the other hand, several groups have reported Diels–Alder reactions in which a phenomenon so-called dynamical bifurcation takes place.4,5 The dynamical bifurcation in the Diels–Alder reaction is the focus of this paper.

Theoretically, an elementary reaction step is defined as a path that connects a pair of two local minima via a single transition state (TS). These paths are commonly defined by the steepest descent path starting from a TS in the mass-weighted coordinates and called intrinsic reaction coordinate (IRC).6,7 By calculating an IRC path, energetic and geometrical variations that occur in the corresponding reaction step can be elucidated.

In most reactions, the IRC path well represents the mechanism of corresponding reaction. However, actual molecular motions deviate from the IRC path due to the kinetic energy, and there are known cases where the neglect of kinetic energy misleads an incorrect mechanism.8–10 The dynamical bifurcation is one of such cases.11–28 In a bifurcation reaction, a single TS relates to two products. In other words, a set of reactive trajectories passing the corresponding TS region branches into two components giving two different products. Such a TS is called an ambimodal TS.5 Since an IRC calculation from a TS gives only a single product, the IRC calculation misses one of two products in bifurcation reactions.29

Occurrence of bifurcation can be recognized by running ab initio molecular dynamics (AIMD) simulations starting from the corresponding TS region.30–53 Therefore, AIMD simulations have been performed to reveal bifurcations in reactions of various types such as organic reactions,30–35,37,38,43–45,47,49–57 organometallic reactions,41,58 and biosynthesis reactions.36,39,59–64 Occurrence of bifurcation can also be discussed through a static analysis of the potential energy surface (PES) by locating a valley-ridge transition (VRT) point along the IRC path. On a VRT point, the shape of PES perpendicular to the IRC path changes from the valley to the ridge.11,12,15,16,20,21,23,24,26,27 To find a VRT point, the curvature of PES needs to be computed at many points along the IRC path.

Our purpose in this paper is to systematically explore TSs from which a dynamical bifurcation takes place. In the pioneering work by Hong and Tantillo,64 they applied AIMD simulations to various TSs in a reaction path network of terpene and discussed bifurcation reactions on the network. To avoid running AIMD simulations from many TSs, Harabuchi et al. systematically explored VRT points by applying the curvature analysis along all IRC paths on a reaction path network of Au5 cluster.65 Then, they applied AIMD simulations to only TSs from which a VRT point was obtained to identify products of corresponding bifurcation reactions. Although this approach allowed them to avoid AIMD simulations from many TSs, the curvature analysis done in their VRT search needs to compute Hessian matrix and is still demanding computationally.

Our approach proposed here was inspired by results in early studies where two different products were obtained for the same TS depending on the choice of coordinate systems.22 Subsequent AIMD studies revealed that a dynamical bifurcation took place from the TS.50–52 In this discovery, use of paths of two different mathematical nature was the key. In other words, both of the two products were identified by computing two static paths with different mathematical nature. We have developed an automated reaction path search method called artificial force induced reaction (AFIR).66,67 The AFIR method traces a path so-called AFIR path because AFIR paths can be computed easily from a local minimum to the other local minimum. In the standard AFIR procedure, all obtained AFIR paths are further processed to obtain actual TSs. Therefore, our idea here is that two bifurcation products could be identified from differences between connections of AFIR paths and IRC paths. It is thus expected that this approach could suggest both occurrence and products of bifurcation reactions without any additional efforts after an automated reaction path search.

In this study, we explored paths of reactions between 2-vinylfuran and 3-methoxycarbonylcyclopentadienone (see Scheme 1) systematically by the multicomponent version of the AFIR method (MC-AFIR).66 These two reactants are known to react to each other affording two Diels–Alder products. Furthermore, paths leading these two products are known to share a single TS and thus correspond to a bifurcation reaction. Moreover, this system has many reactive sites at which different competing Diels–Alder reactions can take place. Actually, the MC-AFIR search generated paths to sixty-five unique products, where these products were unique in terms of their SMILES representation. Among the sixty-five, sixteen corresponded to Diels–Alder products. As expected, we found many cases in which the reactants were linked to a product via only an AFIR path. In these cases, through a careful analysis of search logs of obtaining actual TSs, it was found that two AFIR paths shared a single TS region as the bottleneck. The further curvature analysis of IRC paths from these TSs identified six cases in which VRT points existed along the corresponding IRC paths. Finally, we proposed six bifurcation cases for this single reaction system.


image file: d0cp02379e-s1.tif
Scheme 1 Diels–Alder cycloaddition reaction33 between 2-vinylfuran and 3-methoxycarbonylcyclopentadienone.

II. Theory

The AFIR method induces a structural deformation in a system by an artificial force and finds a path of chemical reaction from a local minimum to the other local minimum. The structural deformation can be induced by minimizing a function so-called AFIR function.66 Thanks to careful design of the form of AFIR function, paths that are obtained through its minimization pass TS regions that actual reaction paths pass.66,68 Therefore, the AFIR method first explores AFIR paths and finds TSs for various reactions by further processing the AFIR paths. Details how it systematically explores AFIR paths and how actual TSs are obtained from AFIR paths are described in our previous papers.66–69

Our idea or hypothesis proposed in this study is explained using a 2-dimensional model PES shown in Fig. 1. On this PES, three local minima, R, P1, and P2, and two TSs, T1 and T2, exist. Let's consider the process of finding paths to P1 and P2 by the AFIR method starting from R. In this case, the AFIR method finds these two products by adding artificial forces along different directions. The corresponding AFIR paths are depicted as white lines. These AFIR paths pass the T1 region, and further geometry optimizations starting from the highest energy points along these AFIR paths converge to T1. In the actual application to the molecular system shown below, an additional procedure of relaxing the AFIR path was taken to avoid failure of TS optimization (see the computational details section for more description). Finally, the IRC path is computed from T1. The IRC path from T1 toward products’ valley once approaches to T2 and finally falls into P1's well.


image file: d0cp02379e-f1.tif
Fig. 1 A schematic PES on which a reaction starting from a reactant MIN, R, reaches two different product MINs, P1 and P2. White lines indicate the AFIR paths, a black line indicates the IRC path, black circles indicate TSs, and a triangle indicate a VRT point.

In this case, two AFIR paths leading two different products provide a common TS. In other words, two AFIR paths share a single TS region. This situation suggests occurrence of the dynamical bifurcation. Actually, the VRT takes place along this IRC path. At T1, the curvature perpendicular to the IRC path is positive. On the other hand, the curvature decreases while approaching T2 and becomes negative around T2. As defined above, the point at which the curvature becomes negative from positive is called a VRT point.

In short, occurrence of dynamical bifurcations can be recognized by collecting cases where AFIR paths to different products provide a common TS. Furthermore, sets of products of these dynamical bifurcations can also be known as products of AFIR paths. This procedure is especially useful when the mechanism of the target reaction is unknown. In such a case, the automated search by the AFIR method is effective because the search systematically finds possible paths and help identifying the most probable mechanism. After the search, one can reveal both occurrence and products of possible dynamical bifurcations just by analyzing search logs without any additional quantum chemical calculations.

III. Results and discussion

The Diels–Alder cycloaddition is a [4+2] cycloaddition reaction between a conjugated diene and dienophile. In this study, a case between 2-vinylfuran and 3-methoxycarbonylcyclopentadienone is studied (see Scheme 1). For this reaction, occurrence of a bifurcation was previously reported.33 The reported bifurcation for this reaction gave two Diels–Alder products; one is the cycloaddition product composed of C3b, C4b, C5b, and C6b in cyclopentadienone and C1a and C2a in 2-vinylfuran (denoted by [C3b–C6b + C1a–C2a] type), and the other is that of C3b and C4b in cyclopentadienone and C1a, C2a, C3a, and C4a in 2-vinylfuran (denoted by [C1a–C4a + C3b–C4b] type). In other words, a cyclopentadienone acts as diene in the case of [C3b–C6b + C1a–C2a] type, while 2-vinylfuran acts as diene in the case of [C1a–C4a + C3b–C4b] type. This type of bifurcation was called a bispericyclic reaction.4 Thus, fragment-A was set to C3b, C4b, C5b, and C6b in cyclopentadienone, and fragment-B is set to as C1a, C2a, C3a, and C4a in 2-vinylfuran.

The systematic AFIR search generated 134 product-MINs and 125 TSs automatically. The search found not only paths of [4+2] cycloaddition but also those of the other types such as [2+2] cycloaddition, [4+4] cycloaddition, [6+2] cycloaddition, and [6+4] cycloaddition. The obtained TSs are ordered in the ascending order of their energies and termed TSx (x = 0–124), where all energy values below are relative to the total energy of reactants. The lowest TS0 is for the [4+2] cycloaddition in Scheme 1. It was found that two AFIR paths shared TS0. These two AFIR paths lead to [C1a–C4a + C3b–C4b] and [C3b–C6b + C1a–C2a] products, respectively. This suggests that TS0 serves as a TS of bifurcation giving [C1a–C4a + C3b–C4b] and [C3b–C6b + C1a–C2a] products. This is consistent with the previous discovery summarized in Scheme 1.33

Fig. 2a compares variations of two internal coordinates along the IRC path from TS0 with those along the two AFIR paths, where the AFIR paths shown in this figure were obtained by relaxing the initial AFIR paths of the low computational level by the final computational level (see computational details). Like the two-dimensional model shown in Fig. 1, the AFIR method found these two products. The corresponding AFIR paths depicted as blue and red lines passed the TS region, and further geometry optimizations starting from the highest energy points along these AFIR paths converged to the TS. The IRC path computed from the TS toward products’ valley once approached to the TS between the two products and finally fell into the [C3b–C6b + C1a–C2a] product.


image file: d0cp02379e-f2.tif
Fig. 2 Analyses of AFIR paths and IRC paths. (a) shows a plot for C2a–C6b distance (x-axis) versus C4a–C4b distance (y-axis) along the IRC paths and the AFIR paths. Blue and red lines indicate AFIR paths, black lines indicate IRC paths, circles indicate TSs, and a triangle indicates a VRT point. Structures of TSs, MINs, and a VRT point are shown. Normal mode vectors corresponding to the negative eigenvalue mode at TSs and a VRT point are depicted. (b) shows the eight lowest vibrational frequencies of the modes perpendicular to the product side of the IRC path from s = 0.0 Å to 1.5 Å for TS0. Red line corresponds to the mode related to the dynamical bifurcation. All electronic energies are shown in kJ mol−1 relative to the set of reactants.

The VRT took place along this IRC path. Fig. 2b shows variation of the eight lowest vibrational frequencies of the modes perpendicular to the IRC path from s = 0.0 Å to 1.5 Å. The negative eigenvalue mode (the red line) at the VRT point (see Fig. 2a) is nearly parallel to the negative eigenvalue mode of the TS between the two products (see Fig. 2a) and thus is related to the bifurcation. These results supported our idea shown in Fig. 1 and stimulated us to further study the other paths with the same idea.

Next, let's consider all [4+2] cycloaddition reactions that are assumable to occur between the diene and dienophile. There are eight combinations of diene and dienophile for [4+2] cycloaddition reactions; [C3b–C6b + C4a–C3a], [C3b–C6b + C3a–C4a], [C1a–C4a + C3b–C4b], [C1a–C4a + C4b–C3b], [C3b–C6b + C1a–C2a], [C3b–C6b + C2a–C1a], [C1a–C4a + C5b–C6b], [C1a–C4a + C6b–C5b]. All the eight patterns have endoexo types, and thus, 16 unique [4+2] cycloadditions are expected in this system. These 16 patterns are listed in Table 1. As shown in Fig. 3a and b, when two dienes approach together during a Diels–Alder cycloaddition, both of [4+2] and [2+4] cycloadditions are expected. This is because both two dienes can act as the dienophile of the reaction. In this case, the [4+2] and [2+4] cycloadditions sometimes pass through a common TS, i.e. bispericyclic reaction4 (indicated by a [4+2]/[2+4] bispericyclic reaction). In Table 1, all the excepted [2+4] reactions which can be paired with the 16 [4+2] reactions as [4+2]/[2+4] bispericyclic reactions discussed above were listed in the row corresponding to each [4+2] cycloaddition reaction. It is mentioned that there are no excepted [4+2]/[2+4] bispericyclic reaction in entry-10, 12, 14, and 16 (indicated by N/A in Table 1). This is because these reactions are exo cycloaddition where dienophiles are terminal part of each molecule, C5b–C6b or C1a–C2a, and a diene part does not approach another diene part during the reaction. In addition, excepted [4+2]/[2+4] bispericyclic reactions of four entries, i.e. entry-1, 3, 5, and 11, are the same as those for entry-7, 13, 9, and 15. This is because both dienes are included in the two reactant molecules, and these were doubly counted. Actually, all the 16 [4+2] cycloaddition products were found by the present AFIR search.

Table 1 All the expected [4+2] cycloaddition reactions related to fragment-A and -B, i.e. C3b–C6b in cyclopentadienone and C1a–C4a in 2-vinylfuran, respectively. Pairs of diene and dienophile, orientations of endo or exo, indices of TSs, and energies of TSs in kJ mol−1 for each cycloaddition are indicated. The cycloadditions which can be paired with the 16 [4+2] cycloadditions as bispericyclic reactions, i.e. [2+4] cycloaddition and [6+4] cycloaddition, are shown in each row. “N/A” indicates a not-applicable case (see text)
Entry [4+2] cycloaddition [2+4] cycloaddition [6+4] cycloaddition
Diene Dienophile ΔETS TS Diene Dienophile ΔETS TS Triene Diene ΔETS TS
a [2+4] and [4+2] cycloadditions of entry-1 correspond to [4+2] and [2+4] cycloadditions of entry-7, respectively. b [2+4] and [4+2] cycloadditions of entry-3 correspond to [4+2] and [2+4] cycloadditions of entry-13, respectively. c [2+4] and [4+2] cycloadditions of entry-5 correspond to [4+2] and [2+4] cycloadditions of entry-9, respectively. d [2+4] and [4+2] cycloadditions of entry-11 correspond to [4+2] and [2+4] cycloadditions of entry-15, respectively.
1a C3b–C6b C4a–C3a endo 40.6 TS 8 C1a–C4a C4b–C3b 35.3 TS 6 C1b–C6b C6a–C3a 68.5 TS 37
2 C3b–C6b C4a–C3a exo 68.2 TS 35 C3a–C6a C6b–C5b 51.4 TS 20 C1b–C3b C1a–C4a 57.7 TS 22
3c C3b–C6b C3a–C4a endo 59.7 TS 26 C1a–C4a C5b–C6b 42.9 TS 10 C1b–C3b C6a–C3a 59.7 TS 26
4 C3b–C6b C3a–C4a exo 50.5 TS 18 C3a–C6a C3b–C4b 44.4 TS 11 C1b–C6b C1a–C4a 85.6 TS 41
5b C1a–C4a C3b–C4b endo 2.6 TS 0 C3b–C6b C1a–C2a 2.6 TS 0 N/A N/A N/A N/A
6 C1a–C4a C3b–C4b exo 23.3 TS 2 C3b–C8b C1a–C2a 23.3 TS 2 N/A N/A N/A N/A
7a C1a–C4a C4b–C3b endo 35.3 TS 6 C3b–C6b C4a–C3a 40.6 TS 8 N/A N/A N/A N/A
8 C1a–C4a C4b–C3b exo 47.5 TS 15 C3b–C8b C4a–C3a 47.5 TS 15 N/A N/A N/A N/A
9b C3b–C6b C1a–C2a endo 2.6 TS 0 C1a–C4a C3b–C4b 2.6 TS 0 N/A N/A N/A N/A
10 C3b–C6b C1a–C2a exo 25.7 TS 5 N/A N/A N/A N/A C1b–C3b C4a–C1a 25.7 TS 5
11d C3b–C6b C2a–C1a endo 24.5 TS 3 C1a–C4a C6b–C5b 24.5 TS 3 N/A N/A N/A N/A
12 C3b–C6b C2a–C1a exo 45.2 TS 12 N/A N/A N/A N/A C1b–C6b C4a–C1a 45.2 TS 12
13c C1a–C4a C5b–C6b endo 42.9 TS 10 C3b–C6b C3a–C4a 59.7 TS 26 N/A N/A N/A N/A
14 C1a–C4a C5b–C6b exo 64.9 TS 34 N/A N/A N/A N/A N/A N/A N/A N/A
15d C1a–C4a C6b–C5b endo 24.5 TS 3 C3b–C6b C2a–C1a 24.5 TS 3 N/A N/A N/A N/A
16 C1a–C4a C6b–C5b exo 63.7 TS 31 N/A N/A N/A N/A N/A N/A N/A N/A



image file: d0cp02379e-f3.tif
Fig. 3 Schematic picture of bispericyclic bifurcations. (a) and (b) are for the case of [4+2]/[2+4] bispericyclic reactions. (c) and (d) are for [4+2]/[6+4] bispericyclic reactions. (e)–(g) are for the case where both of [4+2]/[2+4] bispericyclic reactions or [4+2]/[6+4] bispericyclic reactions are expected.

Similarly, [4+2] and [6+4] cycloadditions sometimes pass through a common TS geometry, when a diene and a triene parts approach together during a [4+2] Diels–Alder cycloaddition like Fig. 3c and d. This type is indicated by a [4+2]/[6+4] bispericyclic reaction, and the expected [6+4] reactions are indicated in Table 1. In the four cases, i.e. entry-1, 2, 3, 4, 10 and 12, there were possibilities of [4+2]/[6+4] bispericyclic reaction. On the other hand, there was no candidate expected for the other eight cases, because a diene in 2-vinylfuran does not approach triene in cyclopentadienone during the reaction of eight cases, which are indicated by N/A in Table 1. Actually, all the expected [6+4] products were also obtained by the present AFIR search.

In the results of the present search, a common TS region was shared by two AFIR paths in seven cases among the 16 bispericyclic reactions listed in Table 1. All the seven cases are indicated by a black flame in Fig. 4. Entry-3, 10 and 12 correspond to [4+2]/[6+4] bispericyclic reactions, and entry-5 (equal to entry-9), 6, 8, and 11 (equal to entry-15) correspond to [4+2]/[2+4] bispericyclic reactions. On the other hand, two different TSs were found for a pair of AFIR paths for [4+2]/[6+4] bispericyclic pair of entry-1, 2, and 4, and for [4+2]/[2+4] bispericyclic pair of entry-1, 2, 3 and 4. In the case of entry-3, [4+2]/[2+4] bispericyclic pair was not found, although that of [4+2]/[6+4] was found. This is explained by the absence of a common chemical bond generated during cycloaddition reactions. As shown in Fig. 3e–g, a common chemical bond is generated during [4+2] and [2+4] cycloadditions, and the situation is same for [4+2] and [6+4] cycloadditions. However, there is no common chemical bond generated during [2+4] and [6+4] cycloadditions, which makes their TSs different. In the cases of entry-1, 2, and 4, three AFIR path had the own TSs. This is explained based on the energy on TSs which connect two product minima. When a bifurcation occurs along a path from reactant-1 (denoted by R1) to product-1 (denoted by P1) and product-2 (denoted by P2), the TS connecting R1 to P1/P2 must be higher than the TS connecting P1 and P2. This is because a dynamical bifurcation leading to P1 and P2 takes place along a decent path from TS between R1 and P1/P2. In other words, a dynamical bifurcation does not occur when the TS between P1 and P2 is higher than the TS between R1 and P1/P2. In the three cases of entry-1, 2, and 3, the TSs between products were higher than the lowest TS which connected the reactant and product. Thus, these are not cases of dynamical bifurcations. In short, there were the seven cases among the 16 bispericyclic reactions where the TS between P1 and P2 was lower than the TS between R1 to P1/P2, and, in the all the seven cases, a common TS region was shared by two AFIR paths.


image file: d0cp02379e-f4.tif
Fig. 4 Expected [4+2] cycloaddition products for the target reaction. All the 16 [4+2] cycloaddition (Diels–Alder) products indicated in Table 1 are shown. The obtained [2+4] and [6+4] cycloaddition products which can be paired with [4+2] cycloadditions as bispericyclic reactions are indicated in the corresponding row. Black frames indicate the cases where two AFIR paths to the products shared a common TS. “N/A” indicates a not-applicable case.

Finally, all the obtained AFIR paths and TSs are summarized. In Fig. 5, energies of all the obtained TSs in the search, i.e. 125 TSs, were plotted against the indices of atom pairs for each TS. Here, ij indicates the atom pair with the shortest atom–atom distance between the two reactants on each TS. Red and green dashed line indicates a TS shared by two or more AFIR paths which gave products with different SMILES representations, and blue dotted line indicates that for the same SMILES representations. Black solid line indicates a TS for one AFIR path. Red dashed lines are for the cases when the VRT corresponding to the bifurcation was found along the IRC path, and green dashed line are for these without the VRT. All the structures for the obtained TSs are shown in the ESI.


image file: d0cp02379e-f5.tif
Fig. 5 A plot of ΔETS against indices of the closest atom pairs on TSs. Black solid lines indicate TSs that one AFIR path passes through it. Blue dotted lines indicate TSs shared by two or more AFIR paths which gave products with the same SMILES representations. Green and red dashed lines indicate TSs shared by two or more AFIR paths which gave products with different SMILES representations. Red dashed lines indicate TSs with a VRT on its IRC path, and green dashed lines indicate TSs without any VRTs.

Interestingly, there were 29 cases where a common TS was shared by two AFIR paths, and there were six cases where a common TS was shared by three AFIR paths. The latter case is related to a trifurcation reported in a previous study.28 Thus, totally 35 candidates of bifurcations were found in the present search. In 16 cases among the 35 cases, two different product minima had the same SMILES representations, which corresponds to the reactions giving conformationally different products. In 19 cases among the 35 cases, two different product minima had different bonding patterns. Among these 19 cases, there was the VRT corresponding to the bifurcation along the IRC path in six cases (indicated by red dashed lines in Fig. 5). The TS geometries and the products of the six cases were depicted in ascending order of TS energies in Fig. 6.


image file: d0cp02379e-f6.tif
Fig. 6 Six cases in which a common TS region was shared by two AFIR paths to reach two different product minima with different SMILES representations. In all the six cases, a VRT was found on the IRC path. Blue dotted line indicates the atom pair which makes the chemical bond during the reaction. TS energies are also indicated in kJ mol−1.

In Fig. 6, in addition to the [4+2]/[2+4] bispericyclic reactions and [4+2]/[6+4] bispericyclic reactions expected in Table 1 (TS0, TS3, TS12, TS15, and TS26), the other one (TS51) was listed. The case of TS51 is a bifurcation to a [2+2] product and a [4+4] product, which has not been known previously. It is emphasized that the prediction of this bifurcation is not easy, but it was automatically obtained by using the present approach without using any prior knowledges of the reaction. These results demonstrate usefulness of the present approach to search for bifurcations.

As shown in Fig. 6, for all the six cases, the TS energies, ΔETS, was lower than 110 kJ mol−1. Also, the six cases correspond to the reaction in which the atom pair was composed by the terminal carbon atoms of a diene, C3b or C6b of cyclopentadienone, and C1a, C3a, C4a, or C6a of 2-vinylfuran. Thus, it is concluded that, in this system, the bifurcations accompany the bond formation between the terminal carbon atoms of the dienes.

IV. Conclusion

In this study, paths of Diels–Alder reactions between 2-vinylfuran and 3-methoxycarbonylcyclopentadienone were systematically explored by the MC-AFIR method. This reaction is known to give two Diels–Alder products from a single TS region through the dynamical bifurcation.33 In this study, we proposed a procedure to systematically predict cases in which the dynamical bifurcation occurs and applied it to the reaction. In this procedure, occurrence of the dynamical bifurcation is identified by collecting cases where AFIR paths to different products share a single TS region. The idea was inspired by an early study in which steepest descent paths computed in different coordinate systems led two different products of the dynamical bifurcation.22

The present MC-AFIR search generated 125 TSs to 65 unique products. In addition to all possible [4+2] cycloaddition paths (Diels–Alder reactions), paths of the other cycloaddition types such as [2+2] cycloaddition, [4+4] cycloaddition, [6+2] cycloaddition, and [6+4] cycloaddition were also found. In 19 among all the 125 TSs, two AFIR paths shared a single TS region as the bottleneck and gave product minima different in terms of SMILES representation. Then, these 19 cases were further studied, and in six cases among the 19, a VRT point was found along the corresponding IRC path. Therefore, these six including the one reported in the literature33 were proposed as TSs causing the dynamical bifurcation. The five new cases corresponded to two [4+2]/[2+4] bispericyclic reactions, two [4+2]/[6+4] bispericyclic reactions, and the other. The present procedure can find TSs causing dynamical bifurcations without using any prior knowledge on the target reaction and thus would be promising in future mechanistic studies on reactions involving dynamical bifurcations.

V. Computational details

The AFIR paths between fragments A and B were computed starting from 1500 random mutual positions and orientations between 2-vinylfuran and 3-methoxycarbonylcyclopentadienone. The maximum model collision energy parameter, γ, was set to 1000 kJ mol−1. The obtained AFIR paths were reoptimized using the LUP method,70 and the energy maxima along the LUP paths were optimized to the actual TSs. From all obtained TSs, the IRC path was computed. These calculations were done at the ωB97X-D71/D95V level. AFIR paths passing the TS0 region were further relaxed by the LUP method at the MPW1K/6-31+G** level to compare the results with those in the literature.33 19 TSs which were shared by two AFIR paths giving products different in terms of SMILES representation were further optimized at the MPW1K/6-31+G** level. The IRC path was calculated for these 19 TSs at the MPW1K/6-31+G** level, and existence of a VRT point along the IRC path was examined through the curvature analysis using Hessian from which components along the gradient vector were eliminated.15 Energy, gradient, and Hessian were computed using the Gaussian 16 program package.72 AFIR, LUP, and IRC calculations were done using the developmental version of GRRM program.73 In this study, a post-processing code which automatically identifies candidates of ambimodal TSs was developed independently to the GRRM program; the code is executed in the directory in which GRRM outputs exist and finds the candidates from the GRRM outputs.

Conflicts of interest

The authors declare no competing financial interest.

Acknowledgements

TI is grateful to the Institute for Quantum Chemical Exploration for the IQCE fellowships for Young Scientists. This study is supported by JST-ERATO with grant number JPMJER1903. SM is supported by JST, CREST with grant number JPMJCR14L5. YH is supported by JST, PRESTO with grant number JPMJPR16N8. We thank Ms Takako Homma for editing a draft of this manuscript.

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Footnote

Electronic supplementary information (ESI) available: (I) Cartesian coordinates of the optimized TS structures at the ωB97X-D/D95V level and (II) variations in the four lowest transverse vibrational mode frequencies along the product side of the IRC path for the TSs which are shared by two AFIR paths. See DOI: 10.1039/d0cp02379e

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